A note on sufficient conditions that a common factor will be determinate in an infinite domain of variables

It is proved for the common factor model withr common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity.The author is indebted to R. P. McDonald for suggesting the proof of Guttman's determinantal equation for the squared multiple correlation in predicting a factor from the observed variables used in the parenthetical note.     

Factor analysis by instrumental variables methods

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.     

群体认同与个体心理健康的关系:调节变量与作用机制

近年来,群体认同与个体心理健康的关系得到了研究者的关注。本研究通过对相关文献的梳理与归纳发现:民族、国家、学校、家庭等不同种类群体的认同,群体认同数量与个体心理健康的关系显著;群体认同与个体心理健康之间的关系受群体认同动机和个体对群体评价的调节,并依赖于自尊、社会支持、控制知觉、归因方式等因素的中介;未来应利用多种方法继续研究群体认同与个体心理健康关系的调节变量与作用机制。     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.     

Mediation analysis and categorical variables: The final frontier

Many scholars are interested in understanding the process by which an independent variable affects a dependent variable, perhaps in part directly and perhaps in part indirectly, occurring through the activation of a mediator. Researchers are facile at testing for mediation when all the variables are continuous, but a definitive answer had been lacking heretofore as to how to analyze the data when the mediator or dependent variable is categorical. This paper describes the problems that arise as well as the potential solutions. In the end, a solution is recommended that is both optimal in its statistical qualities as well as practical and easily implemented: compute z_Mediation.     

Factor indeterminacy in the 1930's and the 1970's some interesting parallels

The issue of factor indeterminacy, and its meaning and significance for factor analysis, has been the subject of considerable debate in recent years. Interestingly, the identical issue was discussed widely in the literature of the late 1920's and early 1930's, but this early discussion was somehow lost or forgotten during the development and popularization of multiple factor analysis. There are strong parallels between the arguments in the early literature, and those which have appeared in recent papers. Here I review the history of this early literature, briefly survey the more recent work, and discuss these parallels where they are especially illuminating.Support by the NRC (Grant No. A4640) and the University of British Columbia (UBC Humanities and Social Sciences Grant 26-9508) is gratefully acknowledged.     

Some relationships between factors and components

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.The author is indebted to anonymous referees for their comments, corrections and suggestions which have led to the improvement of this article.     

Testing for selectivity in the dependence of random variables on external factors

Random variables A and B, whose joint distribution depends on factors (x,y), are selectively influenced by x and y, respectively, if A and B can be represented as functions of, respectively, (x,S_A,C) and (y,S_B,C), where S_A,S_B,C are stochastically independent and do not depend on (x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of A may depend on y. But parameters characterizing stochastic interdependence of A and B, such as their mixed moments, are generally functions of both x and y. We derive two simple necessary conditions for selective dependence of (A,B) on (x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x^′ and y,y^′,

s_xy≤s_xy^′+s_x^′y^′+s_x^′y,     

Simultaneous classification and multidimensional scaling with external information

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or multidimensional scaling and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.     

Best univocal estimates of orthogonal common factors

A general formulation is presented for obtaining conditionally unbiased, univocal common-factor score estimates that have maximum validity for the true orthogonal factor scores. We note that although this expression is formally different from both Bartlett's formulation and Heermann's approximate expression, all three, while developed from very different rationales, yield identical results given that the common-factor model holds for the data. Although the true factor score validities can be raised by a different non-orthogonal transformation of orthogonalized regression estimates—as described by Mulaik—the resulting estimates lose their univocality.     

On the factor score controversy

A summary and interpretation of the recent literature on the indeterminacy of factor scores is given in simple terms. A good index of factor score determinacy is the squared multiple correlation of the factor with the observed variables.     

On the relationship between sources of within- and between-group differences and measurement invariance in the common factor model

Investigating sources of within- and between-group differences and measurement invariance (MI) across groups is fundamental to any meaningful group comparison based on observed test scores. It is shown that by placing certain restrictions on the multigroup confirmatory factor model, it is possible to investigate the hypothesis that within- and between-group differences are due to the same factors. Moreover, the modeling approach clarifies that absence of measurement bias implies common sources of within- and between-group variation. It is shown how the influence of background variables can be incorporated in the model. The advantages of the modeling approach as compared with other commonly used methods for group comparisons is discussed and illustrated by means of an analysis of empirical data.     

Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Mediation with categorical variables compleat

This rejoinder addresses the thoughtful issues raised by the commentators. We hold to the analytical solution of fitting X → M (to obtain a and sa) and X&;M→ (to obtain b and sb) via regression or logistic regression, depending on whether M and Y are continuous or categorical. Then researchers compute za, zb, and ZMediation.     

Gender differences in the relationship of moderator variables to stress and symptoms

Abstract

This study focused on gender differences in the relationship of four moderator variables (hostility, social support, optimism, and Type A behavior) to perceptions of symptoms and stress based on a sample of 120 female and 79 male undergraduates. Results confirmed significant gender differences in hostility, Type A behavior, and social support but not in optimism. Multiple regression analyses exposed distinct gender differences. For females, social support accounted for 21% and hostility 5% of the total variance in perceived stress. For males, only hostility accounted for any significant variance (10%). Hostility contributed 21% and social support contributed 13% to the total variance in the number of symptoms for females. For males, optimism contributed 14% and hostility 5%. Results indicate that social support is a potent variable for females in their perception of both stress and symptoms but essentially unrelated for males. Hostility, however, relates to perceptions of stress and symptoms for both sexes.     

Some Results on Mean Square Error for Factor Score Prediction

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γ _p = Φ ^1/2 Λ _p ′Ψ _p ⁻¹ Λ _p Φ ^1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γ _p or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors. The author is obliged to Maarten Speekenbrink and Peter van Rijn for their assistance with plotting the figures. In addition, I am obliged to the referees for their stimulating remarks.     

The relationship between miscellaneous symptoms and major symptom factors in obsessive-compulsive disorder

The diverse symptomatology of obsessive-compulsive disorder (OCD) is being increasingly regarded as reducible to a few symptom dimensions. However, prevailing factor-analytically derived models of symptom structure omit a number of the well-recognized “miscellaneous” symptoms of OCD. This study sought to determine whether miscellaneous OCD symptoms, ascertained by the Yale-Brown Obsessive-Compulsive Scale symptom checklist, could be differentially and reliably predicted by four symptom factors (obsessions and checking, symmetry and ordering, contamination and cleaning, and hoarding) in two independent groups of individuals with OCD (n = 381 and n = 107). Logistic regression analyses were used to determine the association of each of the miscellaneous symptoms with the symptom factors; then a single confirmatory factor analysis was conducted to test the model of associations in the smaller sample. Sixteen (89%) of the 18 symptoms examined were reliably predicted by one (11 items) or two (5 items) of the factors, with obsessions and checking and symmetry and ordering emerging as foremost predictors. The expanded four-factor model showed good fit with data from the second sample. Results are conceptually meaningful, but suggest the inadequacy of groupings based solely upon overt behaviors. These findings may aid clinical understanding of OCD and be of value to studies using symptom factors to guide investigation of its causes and correlates.     

The role of traffic law enforcements in the relationship between cultural variables and traffic fatality rates across some countries of the world

The aims of the present study were to investigate whether cultural variables are related to law enforcements as well as traffic fatality rates and to examine the role of law enforcements of five risk factors for road safety (i.e., national speed law, national drink-driving law, national motorcycle helmet law, national seat-belt law, and national child restraint law) in the relationship between cultural variables and traffic fatality rates across countries of the world. The aggregated data of the study included Hofstede’s cultural dimensions, Schwartz’s value dimensions, law enforcements of five risk factors for road safety, gross national income per capita, and traffic fatality rates for 97 countries of the world. The results showed that most of the cultural variables were associated with law enforcements of five risk factors for road safety and traffic fatality rates of countries. By bootstrapping, among Hofstede’s cultural dimensions, it was found that the indirect effects of long-term orientation (LTO) on fatality rates (FR) through speed, helmet, and child restraint enforcements were significant, separately. Among Schwartz’s value dimensions, the indirect effects of embeddedness on fatality rates through speed and child restraint enforcements were significant, separately. Intellectual autonomy had also significant indirect effects on fatality rates via speed and child restraint enforcements, separately. Finally, it was found that the indirect effects of affective autonomy on fatality rates through speed and helmet enforcements were significant, separately.